Projective Characters Of Spin Wreath Products

In1911, I. Schur[31]studied the spin character values of the spin symmetric groupSnwhich is a double cover of the symmetric group Sn. He used two important ingre-dients: the frst one later became famously known as the Schur Q-functions and thesecond method is that of Cliford algebras. These symmetric functions play the samerole for the spin symmetric group Snas Schur functions do for the symmetric groupSn. In the context of the theory of McKay correspondence and afne Lie algebras, thefrst ingredient was generalized to all wreath products by the vertex operator calculusin [7] where a large part of irreducible character table were computed. The purposeof this thesis is to generalize the second ingredient and fll up the missing projectivecharacter values of the wreath products.After the classical work of Frobenius and Schur, irreducible characters of the gen-eral wreath products Γn=Γ Snwere constructed by Specht in his dissertation[36].The generalized symmetric groups were also studied by Osima in[27]. Zelevinsky[40]investigated the Hopf algebra structure of the Grothendieck groups for all Γn. Dur-ing the last twenty years there has been a resurgence of activities on the spin group.Stembridge[34]gave a combinatorial defnition of the Schur’s Q-functions, Sergeev[33]found that the hyperoctahedral group has a similar character theory with the sym-metric group, J′ozefak[14]gave a modern account of Schur’s work using superalgebras,Hofman and Humphreys[8]studied the Hopf algebra structure of the spin characters,Nazarov[26]constructed all irreducible representations of the spin group, and Jing[11]provided a vertex operator approach to Schur Q-functions as well as projective char-acter values. Breakthroughs were also made on modular projective representations ofthe symmetric groups in [3] and [17](see also [4]).On the other hand, recognizing the deep connection with the McKay correspon-dence, I. Frenkel, Jing and Wang[7]generalized the frst part of Schur’s work anddetermined all irreducible characters of the spin wreath product Γnof a fnite group Γand the symmetric group Sn. When Γ is a fnite cyclic group, they are double coveringgroups of the generalized symmetric groups, which include hyperoctahedral groups asspecial cases when Γ～=Z2. In Schur’s original work on Sn, the projective characters ofSnare parameterized by strict partitions. Again in the wreath products, the projectiverepresentations are in one to one correspondence to strict partition valued functionson irreducible characters of Γ.It is well-known that the projective characters of Γnjust have nonzero values onthe so-called split conjugacy classes which can be divided into two parts: the evenconjugacy classes corresponding to partition valued functions with odd integer parts and the odd conjugacy classes corresponding to odd partition valued functions withstrict partitions. In [7] the authors determined all irreducible characters of spin wreathproducts by vertex operator calculus and also showed that the character values at allodd conjugacy classes are given by matrix coefcients of products of twisted vertexoperators, thus solved a big chunk of the character table. It seems that the charactervalues on odd strict colored partitions are beyond the reach of vertex operators. Laterin [1] and [25] spin characters for generalized symmetric groups were also consideredusing combinatorial methods and certain basic spin character values were computed.However the character values on odd strict partition valued functions are still unknown,as the method associated with the McKay correspondence and vertex representationsseems not suitable for computing this part of the character table. Knowledge of thiswill be useful in representation theory as they include practically all double coveringsof Weyl groups of classical types.The thesis is organized as follows:In chapter1, we recall some fundamental defnitions and propositions for charactertheory, semisimple superalgebras and supermodules, partition valued functions andsymmetric groups.In chapter2, our purpose is to obtain the missing part of the character table of spinwreath product Γnof the symmetric group with a fnite abelian group Γ. We constructall irreducible characters by certain induced representations of Young subgroups of Γnusing the Mackey-Wigner method of little groups[32]. Then we can determine the spincharacter table of spin wreath products Γnwhen Γ is an abelian group.In chapter3, we complete the spin character table of wreath product of the sym-metric group with any fnite group Γ. The Mackey-Wingner method is failed whenΓ is not the abelian group. So we need to have a new method to determine the spincharacter values. We show that the spin character values are sparsely zero and thenon-zero values are given according to how the partitions are supported on variousconjugacy classes.In chapter4, from the deep relations between the spin symmetric group and thespin hyperoctahedral group, we will consider the missing part of the spin charactertable of the wreath product HΓn.